The word problem in Z 2 and formal language theory

Transcription

1 The word problem in Z 2 and formal language theory Sylvain Salvati INRIA Bordeaux Sud-Ouest Topology and languages June 22-24

2 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

3 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

4 Group languages Group finite presentation: a finite set of generators Σ a finite set of defining equations E

5 Group languages Group finite presentation: a finite set of generators Σ a finite set of defining equations E Word problem: given w in Σ, is w = E 1? Group language: {w Σ w = E 1}

6 Group languages Group finite presentation: a finite set of generators Σ a finite set of defining equations E Word problem: given w in Σ, is w = E 1? Group language: {w Σ w = E 1} the word problem is in general undecidable (Novikov 1955, Boone 1958) the languages of different representation of a group a rationally equivalent relate algebraic properties of groups to language-theoretic properties of their group languages

7 Group languages Group finite presentation: a finite set of generators Σ a finite set of defining equations E Word problem: given w in Σ, is w = E 1? Group language: {w Σ w = E 1} the word problem is in general undecidable (Novikov 1955, Boone 1958) the languages of different representation of a group a rationally equivalent relate algebraic properties of groups to language-theoretic properties of their group languages Example: a group language is context free iff its underlying group is virtually free (Muller Schupp 1983)

8 A simple presentation of Z 2 Generators: {a; a; b; b} Defining equations: a 1 = a, b 1 = b, xy = yx a a b b The associated group language is O 2 = {w {a; a; b; b} w a = w a w b = w b }

9 O 2 and computational group theory Gilman (2005)

10 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

11 MIX MIX = {w {a; b; c} w a = w b = w c } MIX and O 2 are rationally equivalent

12 The Bach language Bach (1981) Wikipedia entry: language

13 The MIX language Marsh (1985) Conjecture: MIX is not an indexed language.

14 MIX and Tree Adjoining Grammars Joshi (1985) [MIX ] represents the extreme case of the degree of free word order permitted in a language. This extreme case is linguistically not relevant. [... ] TAGs also cannot generate this language although for TAGs the proof is not in hand yet.

15 MIX and Tree Adjoining Grammars Vijay Shanker, Weir, Joshi (1991)

16 MIX and mildly context sensitive languages Joshi, Vijay Shanker, Weir (1991)

17 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

18 Original paper

19 A generalization of context-free grammars Rule of a context free grammar: A w 1 B 1... w n B n w n+1 with A, B 1,..., B n non-terminals and w 1,... w n+1 string of terminals.

20 A generalization of context-free grammars Rule of a context free grammar: A w 1 B 1... w n B n w n+1 with A, B 1,..., B n non-terminals and w 1,... w n+1 string of terminals. A bottom-up view: A(w 1 x 1... w n x n w n+1 ) B 1 (x 1 ),..., B n (x n )

21 A generalization of context-free grammars Replace strings by tuple of strings: B(s 1,..., s m ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n )

22 A generalization of context-free grammars Replace strings by tuple of strings: B(s 1,..., s m ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) the strings s i are made of terminals and of the variables x i j,

23 A generalization of context-free grammars Replace strings by tuple of strings: B(s 1,..., s m ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) the strings s i are made of terminals and of the variables x i j, the variables x i j are pairwise distinct (otherwise we get Groenink s Literal Movement Grammars),

24 A generalization of context-free grammars Replace strings by tuple of strings: B(s 1,..., s m ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) the strings s i are made of terminals and of the variables x i j, the variables x i j are pairwise distinct (otherwise we get Groenink s Literal Movement Grammars), each variable x i j has at most one occurrence in the string s 1... s m (otherwise we get Parallel Multiple Context-Free Grammars).

25 Formal definition A m-mcfg(r) is a 4-tuple (N, T, P, S) such that: N is a ranked alphabet of non-terminals of max. rank m.

26 Formal definition A m-mcfg(r) is a 4-tuple (N, T, P, S) such that: N is a ranked alphabet of non-terminals of max. rank m. T is an alphabet of terminals

27 Formal definition A m-mcfg(r) is a 4-tuple (N, T, P, S) such that: N is a ranked alphabet of non-terminals of max. rank m. T is an alphabet of terminals P is a set of rules of the form: where: A(s 1,..., s k ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) A is a non-terminal of rank k, B i is non-terminal of rank k i, n r, the variables x i j are pairwise distinct, the strings si are in (T X ) with X = n ki i=1 j=1 {x j i}, each variable x i j has at most one occurrence in s 1... s k

28 Formal definition A m-mcfg(r) is a 4-tuple (N, T, P, S) such that: N is a ranked alphabet of non-terminals of max. rank m. T is an alphabet of terminals P is a set of rules of the form: where: A(s 1,..., s k ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) A is a non-terminal of rank k, B i is non-terminal of rank k i, n r, the variables x i j are pairwise distinct, the strings si are in (T X ) with X = n ki i=1 j=1 {x j i}, each variable x i j has at most one occurrence in s 1... s k S is a non-terminal of rank 1, the starting symbol.

29 The language generated by an MCFG Given an MCFG G = (N, T, P, S), if the following conditions holds: B 1 (s 1 1,..., s1 k 1 ),..., B n (s n 1,..., sn k n ) are derivable, A(s 1,..., s k ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) is a rule in P then A(t 1,..., t k ) with t i = s i [x i j s i j ] i [1;n],j [1;k i ] is derivable.

30 The language generated by an MCFG Given an MCFG G = (N, T, P, S), if the following conditions holds: B 1 (s 1 1,..., s1 k 1 ),..., B n (s n 1,..., sn k n ) are derivable, A(s 1,..., s k ) B 1 (x 1 1,..., x 1 k 1 ),..., B n (x n 1,..., x n k n ) is a rule in P then A(t 1,..., t k ) with t i = s i [x i j s i j ] i [1;n],j [1;k i ] is derivable. The language define by G, L(G) is: {w S(w) is derivable}

31 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ)

32 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ)

33 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ) Q(c, d)

34 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ) Q(c, d) Q(cc, dd)

35 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ) Q(c, d) Q(cc, dd) P(ɛ, ɛ)

36 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ) Q(c, d) Q(cc, dd) P(ɛ, ɛ) P(a, b)

37 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ) Q(c, d) Q(cc, dd) S(accbdd) P(ɛ, ɛ) P(a, b)

38 An example S(x 1y 1x 2y 2) P(x 1, x 2), Q(y 1, y 2) P(ax 1, bx 2) P(x 1, x 2) P(ɛ, ɛ) Q(cx 1, dx 2) Q(x 1, x 2) Q(ɛ, ɛ) Q(ɛ, ɛ) Q(c, d) Q(cc, dd) S(accbdd) P(ɛ, ɛ) P(a, b) S(a n c m b n d m ) P(a n, b n ), Q(c m, d m ) The language is: {a n c m b n d m n N m N}

39 The well-nestedness constraint I (x 1 y 1, y 2 x 2 ) J(x 1, x 2 ), K(y 1, y 2 ) I (x 1 y 1, x 2 y 2 ) J(x 1, x 2 ), K(y 1, y 2 ) A(x 1 z 1, z 2 x 2 y 1, y 2 y 3 x 3 ) B(x 1, x 2, x 3 ) C(y 1, y 2, y 3 ) D(z 1, z 2 ) A(z 1 x 1, y 1 x 2 z 2 y 2 x 3, y 3 ) B(x 1, x 2, x 3 ) C(y 1, y 2, y 3 ) D(z 1, z 2 )

40 MCFL wn and MCFL MCFL MCFL wn {a n 1... an m n N} {w m+1 w {a; b}, m N}

41 MCFL wn and MCFL MCFL {w 1... w nz nw nz n 1... z 1 w 1 z 0 w r 1... w r n n N, w i {c; d} +, z 0,..., z n D 1 } Staudacher 1993 Michaelis 2005 MCFL wn {a n 1... an m n N} {w m+1 w {a; b}, m N}

42 MCFL wn and MCFL MCFL {w 1... w nz nw nz n 1... z 1 w 1 z 0 w r 1... w r n n N, w i {c; d} +, z 0,..., z n D 1 } Staudacher 1993 Michaelis 2005 MCFL wn {a n 1... an m n N} {w m+1 w {a; b}, m N} {w#w#w w D 1 } Engelfriet, Skyum 1976

43 MCFL wn and MCFL MCFL {w 1... w nz nw nz n 1... z 1 w 1 z 0 w r 1... w r n n N, w i {c; d} +, z 0,..., z n D 1 } Staudacher 1993 Michaelis 2005 MCFL wn {a n 1... an m n N} {w m+1 w {a; b}, m N} {w#w#w w D 1 } Engelfriet, Skyum 1976 {w#w w D 1 } Kanazawa, S. 2010

44 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

45 A 2-MCFG for O 2 S(xy) Inv(x, y) Inv(x 1 y 1, y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 y 1 y 2, x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 x 2 y 1, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, x 2 y 1 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(αx 1 α, x 2 ) Inv(x 1, x 2 ) Inv(αx 1, αx 2 ) Inv(x 1, x 2 ) Inv(αx 1, x 2 α) Inv(x 1, x 2 ) Inv(x 1 α, αx 2 ) Inv(x 1, x 2 ) Inv(x 1 α, x 2 α) Inv(x 1, x 2 ) Inv(x 1, αx 2 α) Inv(x 1, x 2 ) Inv(x 1 y 1 x 2, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(ɛ, ɛ) where α {a; b}

46 A 2-MCFG for O 2 S(xy) Inv(x, y) Inv(x 1 y 1, y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 y 1 y 2, x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 x 2 y 1, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, x 2 y 1 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(αx 1 α, x 2 ) Inv(x 1, x 2 ) Inv(αx 1, αx 2 ) Inv(x 1, x 2 ) Inv(αx 1, x 2 α) Inv(x 1, x 2 ) Inv(x 1 α, αx 2 ) Inv(x 1, x 2 ) Inv(x 1 α, x 2 α) Inv(x 1, x 2 ) Inv(x 1, αx 2 α) Inv(x 1, x 2 ) Inv(x 1 y 1 x 2, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(ɛ, ɛ) well-nested binary rules non well-nested rules where α {a; b}

47 A 2-MCFG for O 2 S(xy) Inv(x, y) Inv(x 1 y 1, y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 y 1 y 2, x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 x 2 y 1, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, x 2 y 1 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(αx 1 α, x 2 ) Inv(x 1, x 2 ) Inv(αx 1, αx 2 ) Inv(x 1, x 2 ) Inv(αx 1, x 2 α) Inv(x 1, x 2 ) Inv(x 1 α, αx 2 ) Inv(x 1, x 2 ) Inv(x 1 α, x 2 α) Inv(x 1, x 2 ) Inv(x 1, αx 2 α) Inv(x 1, x 2 ) Inv(x 1 y 1 x 2, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(ɛ, ɛ) well-nested binary rules rules for constants non well-nested rules where α {a; b}

48 A 2-MCFG for O 2 S(xy) Inv(x, y) Inv(x 1 y 1, y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 y 1 y 2, x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1 x 2 y 1, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, x 2 y 1 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(αx 1 α, x 2 ) Inv(x 1, x 2 ) Inv(αx 1, αx 2 ) Inv(x 1, x 2 ) Inv(αx 1, x 2 α) Inv(x 1, x 2 ) Inv(x 1 α, αx 2 ) Inv(x 1, x 2 ) Inv(x 1 α, x 2 α) Inv(x 1, x 2 ) Inv(x 1, αx 2 α) Inv(x 1, x 2 ) Inv(x 1 y 1 x 2, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(ɛ, ɛ) terminal rule well-nested binary rules rules for constants non well-nested rules initial rule where α {a; b} Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable.

49 A graphical interpretation of O 2. Graphical interpretation of the word aaabaabaabbbbbaabbabbbbaaaabbbbbbbbaaa:

50 A graphical interpretation of O 2. Graphical interpretation of the word aaabaabaabbbbbaabbabbbbaaaabbbbbbbbaaa: The words in O 2 are precisely the words that are represented as closed curves: babbababbabbabbababbaaabbbabbaaaabbabbbaba

51 Parsing with the grammar Rule Inv(ax 1 a, x 2 ) Inv(x 1, x 2 ) Inv(abaabaaababbbabaaabbbabbbbaaaba, babbbbaaaaaababbaab) Inv(baabaaababbbabaaabbbabbbbaaab, babbbbaaaaaababbaab)

52 Parsing with the grammar Rule: Inv(x 1 y 1, y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(baabaaababbbabaaabbbabbbbaaab, babbbbaaaaaababbaab) Inv(bbbabaaabbbabbbbaaab, babbbbaaaaaaba) Inv(baabaaaba, bbaab)

53 Parsing with the grammar Rule Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(bbbabaaabbbabbbbaaab, babbbbaaaaaaba) Inv(bbbabaaabbbabbbbaaab, bbaaaaaa) Inv(babb, ba)

54 Parsing with the grammar Rule: Inv(x 1 b, bx 2 ) Inv(x 1, x 2 ) Inv(bbbabaaabbbabbbbaaab, bbaaaaaa) Inv(bbbabaaabbbabbbbaaa, baaaaaa)

55 Parsing with the grammar Rule: Inv(bx 1, bx 2 ) Inv(x 1, x 2 ) Inv(bbbabaaabbbabbbbaaa, baaaaaa) Inv(bbabaaabbbabbbbaaa, aaaaaa)

56 Parsing with the grammar Rule: Inv(x 1 y 1, y 2 x 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(bbabaaabbbabbbbaaa, aaaaaa) Inv(bbabaaabbbabbbb, aaa) Inv(aaa, aaa)

57 Parsing with the grammar Rule: Inv(bx 1 b, x 2 ) Inv(x 1, x 2 ) Inv(bbabaaabbbabbbb, aaa) Inv(babaaabbbabbb, aaa)

58 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

59 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 1: w 1 or w 2 equal ɛ:

60 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 1: w 1 or w 2 equal ɛ: w.l.o.g., w 1 ɛ, then by induction hypothesis, for any v 1 and v 2 different from ɛ such that w 1 = v 1 v 2, Inv(v 1, v 2 ) is derivable then: Inv(v 1, v 2 ) Inv(ɛ, ɛ) Inv(x 1 x 2, y 1 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(v 1 v 2 = w 1, ɛ)

61 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 2: w 1 = s 1 w 1 s 2 and w 2 = s 3 w 2 s 4 and for i, j {1; 2; 3; 4}, s.t. i j, {s i ; s j } {{a; a}; {b; b}}:

62 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 2: w 1 = s 1 w 1 s 2 and w 2 = s 3 w 2 s 4 and for i, j {1; 2; 3; 4}, s.t. i j, {s i ; s j } {{a; a}; {b; b}}: e.g., if i = 1, j = 2, s 1 = a and s 2 = a then by induction hypothesis Inv(w 1, w 2) is derivable and: Inv(w 1, w 2) Inv(ax 1 a, x 2 ) Inv(x 1, x 2 ) Inv(aw 1 a, w 2)

63 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 3: the curves representing w 1 and w 2 have a non-trivial intersection point:

64 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 3: the curves representing w 1 and w 2 have a non-trivial intersection point: v 1 v 3 A B w 1 w 2 Inv(v 1, v 4 ) Inv(v 2, v 3 ) Inv(v 1 v 2 = w 1, v 3 v 4 = w 2 ) v 4 v 2

65 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 4: the curve representing w 1 or w 2 starts or ends with a loop:

66 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 4: the curve representing w 1 or w 2 starts or ends with a loop: v 1 v 2 Inv(v 1, ɛ) Inv(v 2, w 2 ) Inv(v 1 v 2 = w 1, w 2 )

67 The proof of the Theorem Theorem: Given w 1 and w 2 such that w 1 w 2 O 2, Inv(w 1, w 2 ) is derivable. The proof is done by induction on the lexicographically ordered pairs ( w 1 w 2, max( w 1, w 2 )). There are five cases: Case 5: w 1 and w 2 do not start or end with compatible letters, the curve representing then do not intersect and do not start or end with a loop.

68 Case 5 No rule other than Inv(x 1 y 1 x 2, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) can be used.

69 Case 5 No rule other than Inv(x 1 y 1 x 2, y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) Inv(x 1, y 1 x 2 y 2 ) Inv(x 1, x 2 ), Inv(y 1, y 2 ) can be used.

70 The relevance of case 5 The word abbaabaaabbbbaaaba is not in the language of the grammar only containing the well-nested rules.

71

72

73 The relevance of case 5: a proof is now in hand Joshi (1985) [MIX ] represents the extreme case of the degree of free word order permitted in a language. This extreme case is linguistically not relevant. [... ] TAGs also cannot generate this language although for TAGs the proof is not in hand yet. Theorem (Kanazawa, S. 12) There is no 2-MCFL wn (or TAG) generating MIX or O 2.

74 Solving case 5: towards geometry

75 Solving case 5: towards geometry

76 Solving case 5: towards geometry

77 Solving case 5: towards geometry

78 Solving case 5: a geometrical invariant

79 Solving case 5: a geometrical invariant

80 Solving case 5: a geometrical invariant An invariant on the Jordan curve representing w 1 w 2 : w 1 = aw 1 a and w 2 = aw 2 a w 1 = aw 1 a and w 2 = aw 2 b w 1 = aw 1 a and w 2 = bw 2 a w 1 = aw 1 a and w 2 = bw 2 b

81 Solving case 5: a geometrical invariant An invariant on the Jordan curve representing w 1 w 2 : w 1 = aw 1 b and w 2 = aw 2 b w 1 = aw 1 b and w 2 = bw 2 a w 1 = aw 1 a and w 2 = a w 1 = aw 1 a and w 2 = b

82 Solving case 5: a geometrical invariant An invariant on the Jordan curve representing w 1 w 2 : w 1 = aw 1 b and w 2 = a w 1 = aw 1 b and w 2 = b

83 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

84 Jordan curves illustration from: A combinatorial introduction to topology by Michael Henle (Dover Publications).

85 A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A and D inside J such that AD = A D, then there are two points B and C pairwise distinct from A and D such that A, B, C, and D appear in that order on one of the arcs going from A to D and AD = BC. A D A D

86 A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A and D inside J such that AD = A D, then there are two points B and C pairwise distinct from A and D such that A, B, C, and D appear in that order on one of the arcs going from A to D and AD = BC. A D A D

87 A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A and D inside J such that AD = A D, then there are two points B and C pairwise distinct from A and D such that A, B, C, and D appear in that order on one of the arcs going from A to D and AD = BC. B C A D A D F E G H

88 A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A and D inside J such that AD = A D, then there are two points B and C pairwise distinct from A and D such that A, B, C, and D appear in that order on one of the arcs going from A to D and AD = BC. Applying this Theorem solves case 5.

89 A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A and D inside J such that AD = A D, then there are two points B and C pairwise distinct from A and D such that A, B, C, and D appear in that order on one of the arcs going from A to D and AD = BC. Applying this Theorem solves case 5.

90 A theorem on Jordan curves Theorem: If A and D are two points on a Jordan curve J such that there are two points A and D inside J such that AD = A D, then there are two points B and C pairwise distinct from A and D such that A, B, C, and D appear in that order on one of the arcs going from A to D and AD = BC. Applying this Theorem solves case 5.

91 Winding number Let wn(j, z) be the winding number of a closed curve around z. illustration from: A combinatorial introduction to topology by Michael Henle (Dover Publications).

92 An interesting Lemma Let exp : { C C {0} z e 2iπz. Lemma Given an simple arc AB such that AB = k N, we have: wn(exp( AB), 0) = k

93 Translation becomes rotation { C C {0} exp : z e 2iπz. B C B, C J I A, D A D I, J G, H O E, F F E G H

94 Translation becomes rotation { C C {0} exp : z e 2iπz. B C B, C J I A, D A D I, J O E, F F E

95 An interesting characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve.

96 Jordan curves and winding numbers illustration from: A combinatorial introduction to topology by Michael Henle (Dover Publications). Theorem: There is k { 1; 1} such that the winding number of Jordan curve around a point in its interior is k, its winding number around a point in its exterior is 0.

97 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof

98 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof by 1-periodicity of exp, if then exp( AD) is not a Jordan curve, AD contains a proper subarc BC such that AD = BC,

99 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof by 1-periodicity of exp, if then exp( AD) is not a Jordan curve, AD contains a proper subarc BC such that AD = BC, if exp( AD) is not a Jordan curve: take the closed curve C obtained by removing the closed subcurves of exp( AD) that have a negative winding number,

100 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof by 1-periodicity of exp, if then exp( AD) is not a Jordan curve, AD contains a proper subarc BC such that AD = BC, if exp( AD) is not a Jordan curve: take the closed curve C obtained by removing the closed subcurves of exp( AD) that have a negative winding number, take a proper closed subcurve D of C that is minimal for inclusion,

101 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof by 1-periodicity of exp, if then exp( AD) is not a Jordan curve, AD contains a proper subarc BC such that AD = BC, if exp( AD) is not a Jordan curve: take the closed curve C obtained by removing the closed subcurves of exp( AD) that have a negative winding number, take a proper closed subcurve D of C that is minimal for inclusion, D is a Jordan curve winding positively (i.e. once) around 0,

102 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof by 1-periodicity of exp, if then exp( AD) is not a Jordan curve, AD contains a proper subarc BC such that AD = BC, if exp( AD) is not a Jordan curve: take the closed curve C obtained by removing the closed subcurves of exp( AD) that have a negative winding number, take a proper closed subcurve D of C that is minimal for inclusion, D is a Jordan curve winding positively (i.e. once) around 0, D induces a proper subcurve J of exp( AD) whose winding number is 1,

103 Proving the characterization Lemma Given an simple arc AD such that AD = 1, we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Proof by 1-periodicity of exp, if then exp( AD) is not a Jordan curve, AD contains a proper subarc BC such that AD = BC, if exp( AD) is not a Jordan curve: take the closed curve C obtained by removing the closed subcurves of exp( AD) that have a negative winding number, take a proper closed subcurve D of C that is minimal for inclusion, D is a Jordan curve winding positively (i.e. once) around 0, D induces a proper subcurve J of exp( AD) whose winding number is 1, J induces a proper subarc BC of exp( AD) such that AD = BC.

104 The characterization on the example { C C {0} exp : z e 2iπz. B C B, C J I A, D A D I, J G, H O E, F F E G H

105 The characterization on the example { C C {0} exp : z e 2iπz. B C B, C J I A, D A D I, J O E, F F E

106 Yet another observation from algebraic topology Let s suppose that { AD = 1 and that A 0 = A = 0, A 1 = D = 1,..., A k = k C C {0} let exp : z e 2iπz. B C B, C J I A, D A 2 A 1 A0 A1 A2 A3 A D I, J G, H O E, F A0, A1 F E G H exp sums up the winding number of a Jordan curve around the A i s as the winding number around exp(a 0 ) = exp(0) = 1.

107 Proving the Theorem Let s suppose that AD = 1, Lemma Given an simple arc AD we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve.

108 Proving the Theorem Let s suppose that AD = 1, Lemma Given an simple arc AD we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Corollary: a simple path J from A to D (resp. D to A) does not contain B and C as required in the Theorem iff ϕ(j) is a Jordan curve of C {1} that winding 0 or 1 (resp. or 1) time around 1.

109 Proving the Theorem Let s suppose that AD = 1, Lemma Given an simple arc AD we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Corollary: a simple path J from A to D (resp. D to A) does not contain B and C as required in the Theorem iff ϕ(j) is a Jordan curve of C {1} that winding 0 or 1 (resp. or 1) time around 1. Corollary: if J is a simple closed curve of C composed with two curves J 1 and J 2 respectively going from A to D and D to A which do not contain points B and C as required in the Theorem then wn(exp(j), 1) = wn(exp(j 1 ), 1) + wn(ϕ(j 2 ), 1) 1.

110 Proving the Theorem Let s suppose that AD = 1, Lemma Given an simple arc AD we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Corollary: a simple path J from A to D (resp. D to A) does not contain B and C as required in the Theorem iff ϕ(j) is a Jordan curve of C {1} that winding 0 or 1 (resp. or 1) time around 1. Corollary: if J is a simple closed curve of C composed with two curves J 1 and J 2 respectively going from A to D and D to A which do not contain points B and C as required in the Theorem then wn(exp(j), 1) = wn(exp(j 1 ), 1) + wn(ϕ(j 2 ), 1) 1. Lemma: if J is a simple closed curve of C composed with two curves J 1 and J 2 respectively going from A to D and D to A such that 0 and 1 are in the interior of J, then wn(ϕ(j), 1) 2.

111 Proving the Theorem Let s suppose that AD = 1, Lemma Given an simple arc AD we have: AD contains a proper subarc BC such that AD = BC iff exp( AD) is not a Jordan curve. Corollary: a simple path J from A to D (resp. D to A) does not contain B and C as required in the Theorem iff ϕ(j) is a Jordan curve of C {1} that winding 0 or 1 (resp. or 1) time around 1. Corollary: if J is a simple closed curve of C composed with two curves J 1 and J 2 respectively going from A to D and D to A which do not contain points B and C as required in the Theorem then wn(exp(j), 1) = wn(exp(j 1 ), 1) + wn(ϕ(j 2 ), 1) 1. Lemma: if J is a simple closed curve of C composed with two curves J 1 and J 2 respectively going from A to D and D to A such that 0 and 1 are in the interior of J, then wn(ϕ(j), 1) 2. The Theorem follows by contradiction.

112 Outline The group language of Z 2 A similar problem in computational linguistics Multiple Context Free Grammars (MCFGs) A grammar for O 2 Proof of the Theorem A Theorem on Jordan curves Conjectures

113 Nederhof s conjecture Nederhof (2016) Conjecture: for every k, O k = {w {a 1, a 1..., a k, a k } 1 i k, w ai = w ai } is generated by the grammar with rules of the form: S(x 1... x k ) Inv(x 1,..., x k ) Inv(s 1,..., s k ) Inv(x 1,..., x k ), Inv(y 1,..., y k ) s 1... s k perm(x 1... x k y 1... y k ) Inv(x 1,..., αx i,..., αx j,..., x k ) Inv(x 1,..., x k ). Inv(ɛ,..., ɛ)

114 Status of the conjecture Positive arguments The conjecture has been tested on millions of examples

115 Status of the conjecture Positive arguments The conjecture has been tested on millions of examples In the case of Z 3, some cases can be solved using braiding arguments Negative argument

116 Status of the conjecture Positive arguments The conjecture has been tested on millions of examples In the case of Z 3, some cases can be solved using braiding arguments Negative argument for the case of Z 2 many arguments are strongly related to planarity no clear way of generalizing to higher dimensions